Measure space explained

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

A measurable space consists of the first two components without a specific measure.

Definition

A measure space is a triple

(X, l A, \mu), whereX is a setl A is a -algebra on the set X \mu is a measure on (X, l{A}) (X, l{A}) together with a measure on it.

Example

Set

X = \{ 0 , 1 \} . The $\backslash sigma$-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by $\backslash wp(\backslash cdot).$ Sticking with this convention, we set$$\backslash mathcal\; =\; \backslash wp(X)$$

In this simple case, the power set can be written down explicitly:$$\backslash wp(X)\; =\; \backslash .$$

As the measure, define $\backslash mu$ by $$\backslash mu(\backslash )\; =\; \backslash mu(\backslash )\; =\; \backslash frac,$$so $\backslash mu(X)\; =\; 1$ (by additivity of measures) and $\backslash mu(\backslash varnothing)\; =\; 0$ (by definition of measures).

This leads to the measure space $(X,\; \backslash wp(X),\; \backslash mu).$ It is a probability space, since $\backslash mu(X)\; =\; 1.$ The measure $\backslash mu$ corresponds to the Bernoulli distribution with $p\; =\; \backslash frac,$ which is for example used to model a fair coin flip.

Important classes of measure spaces

Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

• Probability spaces, a measure space where the measure is a probability measure
• Finite measure spaces, where the measure is a finite measure

\sigma -finite measure spaces, where the measure is a \sigma -finite measure

Another class of measure spaces are the complete measure spaces.

References

^{[1] [2] [3]}

Notes and References

1. Book: Kosorok . Michael R. . 2008 . Introduction to Empirical Processes and Semiparametric Inference . New York . Springer . 83. 978-0-387-74977-8 .
2. Book: Klenke . Achim . 2008 . Probability Theory . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 18.
3. Book: Klenke . Achim . 2008 . Probability Theory . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 33.